#413129

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,11,16)(7,17,21,18)(8,10,15,12)(9,23,25,22)(13,27,29,20)(14,24,30,26)(19,33,35,28)(31,36,32,34);(1,3,4,2)(5,7,11,21)(6,13,16,29)(8,18,15,17)(9,19,25,35)(10,27,12,20)(14,28,30,33)(22,32,23,31)(24,36,26,34);(2,3)(5,8,10,16,11,15,12,6)(7,18,20,13,21,17,27,29)(9,22,24,30,25,23,26,14)(19,32,34,28,35,31,36,33);(5,9,12,26,11,25,10,24)(6,14,15,23,16,30,8,22)(7,19,20,34,21,35,27,36)(13,28,17,31,29,33,18,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320325);